On Jan 26 2006, Craig wrote:
> Here's a slider I'd like to see, just for fun (and I have almost
> succeeded in creating one in Geometer's Sketchpad):
> an "eccentricity" slider. Slide from 0 through 3, say, and see a
> conic section change from a circle, through ellipses, to a parabola,
> then through hyperbolas. I can get the effect by using overhead
> transparencies with parallel lines and/or concentric circles...

Just enter the appropriate family of equations
(eg (1-e^2)x^2+y^2=1 if you don't mind the intermediate case being
parallel lines rather than a parabola) and adjust the slider for e.

Or if you want to be more general you could enter
Ax^2+By^2+Cx+Dy+Fxy=1 and vary any of the parameters to your heart's
content.

These implicit forms get a bit ragged as you move the slider because recomputing
an implicit plot fast enough to animate is hard to do without making some
tradeoffs, but if you just click on the equation and press 'enter' again when
you have got to a case that you want to see more neatly then that should clean
it up.

Alternatively, you could use the polar form
r=1/(1+ecos(theta))
(which does give you the parabola as the intermediate case if that is what you
want)and in this case the graphs should stay reasonably nice looking throughout
the animation.

This material is based upon work supported by the
National Science Foundation under Grant DUE-0226284.
Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s)
and do not necessarily reflect the views of the
National Science Foundation.